tau-vee convolution an alternative to the “sliding function” method of convolution
Post on 21-Dec-2015
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Tau-Vee Convolution
An alternative to the “sliding function” method of convolution
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Contents
• What is Convolution (Slides 3-5)
• Preliminaries (Slides 6-7)
• A Detailed Example (Slides 8-47)
• Additional Examples (Slides 48-62)
• Summary (Slides 63-77)
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Convolution with Impulses
... 0 0 ...y t x h t x h t
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Approximating Continuous Data
0.5
0.5
ˆk
k
x k x t dt x k
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x t
t
t
h t
x
x
h t
Overlap
0 0
ˆ
k
k
y t x h t x h t
x k h t k
y t x k h t k x h t d
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Disclaimer
• This presentation is free, without any restrictions, to anyone who wants to use it.
• There is no copyright on this presentation.
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What you will need
• A pencil (possibly with an eraser if you make mistakes).
• A printer because you may find the presentation easier to follow if you print out a few slides.
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Let’s get started
Let d where
0 1 0 0
1 1 2 0 1 and
2 2 3 0 1
0 3
Find .
y t f t g t f g t
t t
t t tf t g t
t t
t
y t
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Change of variables
Define so that
0 1 0 0
1 1 2 0 1 and
2 2 3 0 1
0 3
v t
v
v vf g v
v
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Breakpoints
The points where the definitions of and change
are called breakpoints. Breakpoints occur at 1,2,3
and 0,1 . As shown in the next slide, these breakpoints
divide the plane into a number of
f g v
v
v
subdomains. In the
next series of slides you will be guided through the process of
constructing a map of these subdomains. If you would like
to follow along with this part of the presentation please
begin by printing a copy of the next slide.
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Print this slide
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Follow Along if You Dare
Copy the next series of lines and points onto your printed slide.
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Done!
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So what?
You have now divided the plane into 6 diagonal bands with each band corresponding to one range of time:
Band 1: t less than -1
Band 2: t between -1 and 0
Band 3: t between 0 and 2
Band 4: t between 2 and 3
Band 5: t between 3 and 4
Band 6: t greater than 4
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Band 1:Tau-Vee Method
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Band 1: Sliding Function Method
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Band 2:Tau-Vee Method
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Band 2: Sliding Function Method
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Band 3:Tau-Vee Method
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Band 3: Sliding Function Method
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Band 4: Tau-Vee Method
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Band 4: Sliding Function Method
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Band 5: Tau-Vee Method
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Band 5: Sliding Function Method
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Band 6:Tau-Vee Method
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Band 6: Sliding Function Method
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Further Comments
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Question: Do these drawings have to be very carefully drawn in order to work?
Answer: Probably as long as you have the breakpoints in order it will still work. The next slide shows a pretty messy free-hand version of the diagram for the same example.
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Another fine mess
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This drawing was pretty crummy when I drew it. To make it worse, I spilled weak coffee on it , rode over it with a bicycle, crumpled it up, uncrumpled it, and stomped it on the ground. Only then did I scan it into this presentation. You can still see the 6 diagonal bands and should be able to figure out the integrands and integration limits for all 6 bands! It is still usable! Amazing!
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I also deliberately did not use the correct scale. You can see that the distance from tau=-1 to tau=2 is almost equal to the distance between tau=2 and tau=3. That’s part of the reason why the lines of constant t are deformed into curves instead of nice straight lines at a 45 degree angle. Just try to avoid “time catastrophes”—two different constant time lines intersecting!
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Can this method work for discrete convolution?
Let
1,2, 1, 3 for 0,1,2,3
0 for all other values of
2,4,6 for 5,6,7
0 for all other values of
Let m
nx n
n
ny n
n
z n x n y n x m y n m
k n m
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Therefore
2,8,12,2, 18, 18 for 5,6,7,8,9,19
0 for all other values of
nz
n
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Reference
The tau-vee method for discrete convolution is essentially identical to a method previously described in the following reference:
Enders A. Robinson, “The Minimum Delay Concept in System Design, Part I”, Digital Electronics, Dec. 1963.
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Functions that Extend to Infinity
All functions in the previous examples were nonzero only over a finite interval of time. Does this method work for functions that start and or stop at infinity?
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Example
Let
for 0 and for 1
1 for 0 1
and let 0 for all 1 while g 0 whenever t 1.
Note that the definition of extends back towards .
Find *
te tf t g t t t t
t t
f t t t
f t
f g
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Summary of Tau-Vee Method
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Lines of constant v
Breakpoints in are represented by
horizontal lines.
g v
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Lines of constant v
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Lines of constant tau
Breakpoints in are represented by
vertical lines.
f
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Lines of constant tau
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Corners
Corners occur at each intersection of a constant
line with a constant line. The value of at a corner
is given by
v
t
t v
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Corners
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Lines of Constant t
Lines of constant are diagonal lines (or curves) along which
the value of is constant. Lines of constant are drawn
through each corner. These lines then divide the plane into
a number of d
t
t v t
v
iagonal bands.
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Lines of Constant t
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Integration Path
Each band corresponds to a range of values of . The
convolution within a band is represented by a path on
the diagram. For the convolution of piecewise defined
functions, each time the path crosses a
t
line of constant
the definition of changes. Each time the path
crosses a line of constant the definition of changes.
v g v
f
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Integration Path
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Integration Path
1 2
2 2 2 1
For values of in this band:
*
t
f g f g v d
f g v d f g v d
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Integration limits
When the path crosses a line of constant , the
corresponding integration limit is simply the value of
. When the path crosses a line of constant the
corresponding integration limit is found by sol
v
ving
the equation for the value of , i.e. .
In this case the integration limit will be in the form of
a number. For example:
v t t v
t
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Integration Limits
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Eliminating the variable, v
6
1 2
?
3 ?
2 2 2 1
6 3
*
? Depends on intersections not shown on the
previous diagram
t
t
f g f g t d
f g v d f g v d